The Wikipedia summary of the case suggests that the ruling was much more than it was. Courts tend to rule on narrow issues, as it did here.
The question before the Court was not whether a domain name is a form of property. Both parties agreed that a domain name are a form of property; that wasn't in dispute. The Court did affirm their position in black letter law, though.
Rather, the question before the Court was whether, as intangible property, a tort of conversion claim could be made against it under California law. Previously, cases disagreed as to whether it could be recognized. The Court found that it could be.
https://en.wikipedia.org/wiki/Kremen_v._Cohen
but it's tricky because the domain name registries and registrars have a kind of thicket of contracts nowadays that subordinates registrants' property interests to other rules and considerations. So I don't think a court would think that a modern-day domain name registrant has property rights in a domain name that supersede all of those contractual terms.
Edit: my recollection is that Kremen v. Cohen interpreted the legal defaults that applied because Kremen registered the domain name in an era when Network Solutions didn't require registrants to agree to a contract that limited Network Solutions's liability and gave it various explicit rights and powers.
I'm not a huge ICANN expert, but I think ICANN broadly thinks the status quo is preferable to one in which registrants had stronger, less equivocal property interests in names.